JHEP07(2008)103 July 9, 2008 July 24, 2008 April 15, 2008 ries. uble-trace de- Accepted: Published: Received: eement with previously lacians shows up in the eld at certain values of Poincare metrics of the volves a quantum 1-loop accounts for a sub-leading he Q-curvature: the main nders the universal type A oundary metrics reduces the al anomaly, and its conformal variant powers of the Laplacian a for the type A anomaly coeffi- cerning GJMS operators, to our Published by Institute of Physics Publishing for SISSA AdS-CFT Correspondence, Anomalies in Field and String Theo We argue that the AdS/CFT calculational prescription for do [email protected] ubltUiesta zuHumboldt-Universit¨at Berlin, Institut Physik, f¨ur Newtonstr.15, D-12489 Berlin, Germany E-mail: formations leads to aprimitive, holographic associated derivation to of the the whole(GJMS conform family operators) of at conformally co thecorrection conformal to boundary. the SUGRA The actionterm bulk and in the side the boundary in large-N counterpart two-point limit. function The of sequence theits of CFT scaling GJMS operator dimension. conformal dual Lap bulk The to computation restriction to a to that bulkanomaly. conformally of scalar flat volume In fi renormalization b this whichterm way, re we in directly Polyakov formulas connectFefferman-Graham on two construction, one chief on roles hand, theconjectured of and other patterns t its hand. including a relation generic We tocient and find that simple the matches agr formul all reportedknowledge. values in the literature con Keywords: Abstract: Danilo E. D´ıaz Polyakov formulas for GJMS operators from AdS/CFT JHEP07(2008)103 6 6 7 8 1 13 14 11 14 17 19 early part of last century. tions 5 ribing scalar curvature on a al Laplacian, is best known ave operator, both in curved ry of conformal invariance of subject of a continuous inter- proof of the conformal invariance – 1 – 5.3 Conformal primitive: Polyakov formulas 5.1 The infinitesimal5.2 variation: conformal Type anomaly A holographic anomaly 13 2.1 The generalized prescription for double-trace deforma 2.2 Bulk one-loop2.3 effective actions Boundary partition2.4 function The functional determinants 6.1 Further data for the conformal Laplacian 16 3. Digression on regularizations 4. The general case of5. Poincare metrics Conformally flat class and volume renormalization 12 1. Introduction Conformally covariant differential operators have been the play between physics and mathematicsMaxwell’s ever equations since by the Cunningham discove [1]To and this early Bateman list [2] belongs in theof the Dirac the operator, massless after Dirac Pauli’s equationspacetime. [3], The as Riemannian well as variantto the of mathematicians conformal the for w its later, roleRiemannian the in manifold conform the [4]. Yamabe problem of presc 2. The physical motivation: AdS/CFT correspondence 5 Contents 1. Introduction 6. Comparison to “experiment” A. GJMS operators and Q-curvature B. Q-curvature and volume renormalizationC. Rigid case in dimensional regularization 20 22 7. Conclusion JHEP07(2008)103 , s k . In (1.2) but it (1.1b) in any 3 g w 2 e . the quotient = , associated to b g jecture by Deser re, in general even per is the conformal were obtained using in two dimensions, k nd by Paneitz [5], and 2 related to the null space (or a power thereof and + (global term) (1.1a) , with leading term ∆ P k n and Sparling [9] further  A r generalize to a family of ss, the pattern is preserved auge fixing Maxwell equa- g affian. +(global term) e systematic construction of se by case in low dimensions antly with the dimension of  nomaly) as a combination of y Riegert [7] while pursuing a on whose conformal variation g e GJMS operators via analytic . These ‘conformally covariant v formula [8] for the conformal ator F dv k is, whereas the universal part is 2 conformal primitive − F dv A (generalized) Polyakov formula ˆ g the conformally flat class, where the Weyl ≥ ally related metrics rnative route. − ˆ g w , d tional determinant is usually meant. d b F dv P b of even order 2  F dv +  k M 2 d Z M P Q transformation law under conformal rescaling Z + – 2 – encoded in a = +  g 2 d g b Q dv dv linear d  dw w Q e d + P ˆ g 1 2 ), a local conformal invariant and a total derivative; + dv d of the manifold is odd or d b . However, Branson [13] succeeded in finding a pattern in term Q Q d   M Pfaffian w w 1 M M , in the conformally flat case, to rewrite ‘more invariantly’ Z Z : c c d =2 = b A A 6 and conjectured to hold in all even dimension. A related con , of the Q-curvature generalizes that of the scalar curvature 4 , g det det stands for a local curvature invariant and the global term is Q-curvature w 2 F log = 2 In the early 80’s a fourth-order conformal covariant was fou The feature of the GJMS operators that we will treat in this pa It was in this context that Branson first defined the Q-curvatu e In a physical setting, see the recent work [11, 12] for an alte In the mathematical literature, the zeta-regularized func See [15] and [16] for independent proofs. The restriction to − d 1 2 3 = b of the this family of operators.from These formulas heat can kernel be coefficients workedthe out whose compact ca complexity manifold grows signific with suitable positive ellipticity properties) at conform of functional determinants of a conformally covariant oper even dimension other words, we focus on the conformal anomaly, and its tensor vanishes, was anticipated in [17]. powers of the Laplacian’ (“GJMS” operators in what follows) independently by Eastwood andtions Singer respecting conformal [6], symmetry. indifferent It relation goal, was namely with rediscovered a g b four-dimensional(or analog trace, of or Polyako Weyl)leads anomaly, to i.e. the a general formshowed non-local of that covariant the the acti anomaly. conformal Graham,conformally Laplacian Jenne, covariant and Maso differential the operators Paneitz operato variation of their functional determinant the Fefferman-Graham ambient metric [10],conformal a invariants. chief tool for th Here whenever the dimension and Schwimmer [14] expressesthe the Euler infinitesimal density (or variation (a (kernel) of the operator;captured both by vary the depending Q-curvature on term.in what Away from conformal flatne g can be rephrased in terms of the Q-curvature insteaddimension, of from the the Pf zeroth ordercontinuation (in in derivatives) the term of dimension. th The JHEP07(2008)103 d of S ) at λ or ( d ) is the M λ R S ( holographic M on S λ [23]. A further associated to a ) volume. A key xpansion. onstant regimes in e arises in the volume results in a lap; an underlying non- of the log-divergent term am [10], as an outgrowth n addition to the purely aplacian that arise in the L triggered by the AdS/CFT ncare metrics associated to attering operator bulk metric odel of the hyperbolic space e Lagrangian factorizes and ried out by Henningson and esults originally discussed in rman-Graham construction. ators and to Q-curvature [25, rs, as outlined in [29, 30]. In r the observation by Graham refore, of the Q-curvature) in omaly of the gauge theory on perbolic space. They are also bachevsky space) and confor- es the geometrical task in the dary metric, for certain non- rgences in the bulk are related onformally flat manifolds. eatured by the classical super- the conformal representative of nstein manifolds [18], i.e. man- conformal anomalies which show infrared divergent IR-UV connection ( is also proportional to the integral of of conformal dimension L λ matter O – 3 – associated to the volume is given by the coefficient supports the successful matching in two dimensions. The N . The “rigid” version of the scattering operator of a CFT operator N (rank of the gauge group) and large ’t Hooft coupling, to the at the conformal infinity [22] and the subsequent evaluation i λ ∈ N O k λ , k O holographic anomaly h 2+ d/ = λ for the Q-curvature in terms of the coefficients of the volume e conformal structure , respectively. Another context in which the Q-curvature plays a central rol These later developments involve scattering theory for Poi In the mapping of anomalies at leading large N, the coupling c of the volume expansion and its integral gives the coefficient ) +1 d d ( the Einstein action with alimit negative in cosmological which term the becom gravity effective approximation. description of When string thethe theory challenge bulk consists is metric in f the is regularization Einstein, of the th infinite given feature of the AdS/CFT dualityto is the ultraviolet fact ones that on infrared the dive boundary theory, the so called renormalization of conformally compactifolds asymptotically whose Ei filling metricThe is renewed interest the in Poincare theof metric seminal the of work relation of the between Fefferman the Feffe mal and geometry geometry Grah of on hyperbolic the space sphereCorrespondence (Lo at in the conformal physics infinity, [19 has – 21]. been The reconstruction of a and integrates to a multiple of the Euler characteristic on c the metric at the boundary.Skenderis [24]. This The analysis was thoroughly car failure of the renormalized bulk action to be independent of elaboration thereof leads to thethe mapping boundary, of at the conformal large an up in thegeometric trace terms of involving the curvaturetrivial holographic invariants profiles energy-momentum of of tensor, the therigid boun i matter case [28 fields. – 30]particular, naturally Here, the generalize the GJMS to powers Laplacians the show of GJMS up operato the as L residues of the sc the conformal structure as an27]. alternative route They to are GJMS influenced oper by,the and physical at context the by same Witten timeclosely [21] generalize, related r for to the “rigid” the case AdS/CFT of computation hy of two-point function the Q-curvature. A further refinement by Graham and Juhl [26] v in the volume asymptotics.and The Q-curvature Zworski enters [25] here that afte the “integrated anomaly” formula the poles which the bulk andrenormalization boundary of computations the are coefficient done ofthe do the anomaly Euler not at term over leading (and, large the as conformal boundary of theH half-space model or of the ball m JHEP07(2008)103 ∈ de- k uni- (1.3) , 8 k eading double- 2+ CF T torizes and d/ = λ renders then the ]) as its conformal vature does agree. to g M ; [ λ . M motivation for the func- ) , let alone the full deter- is based on a remarkable λ d th infinitesimal and finite our dimensions, where the ly for the Weyl terms [31]. maly, which is then traced ( The usual Hadamard regu- eneral situation will involve on of the Polyakov formulas ed to individual differential only restriction of conformal M ding Polyakov formula which S/CFT prescription to treat he rigid situation [37]: y. We then review the rigid tiplet. However, in six dimen- remains (isometric to) the hy- lization of the formal equality he crucial point that simplifies d” case of hyperbolic space as S aturally arise in both contexts. n of functional determinants of it computations are then pre- ed and agreement between the in the general case of a filling rmally flat class, i.e. metrics on sion et, via a holographic procedure are evaluated using the Green’s X n on the boundary, but the zation techniques. Next we state functions due to a relevant X log det − = , the bulk d S )] )] on the compact boundary for the ‘+branch’ and its analytic continuation λ λ λ – 4 – M − − S d d ( ( λ λ − − X X [∆ [∆ + − det det log holographic conformal anomaly [32] is still waiting for a [22, 38]. In this case, the volume of the hyperbolic space fac 9 of the CFT, namely a quantum 1-loop in the bulk and a next-to-l branch’. The continuation in the spectral parameter +1 d − AdS H +1 dimensional manifold with a Poincare metric and ( d for the ‘ type A anomalous term associated to the Pfaffian or to the Q-cur ) as a λ + In this note, we propose a heuristic “holographic” derivati The paper is organized as follows: we start with the physical − g conformal to the standard round metric on ; d as argument of the scattering operator X contribution in the large-N( expansion at the boundary. The g minant, and to directly connect the Q-curvature terms that n for the GJMS operatorsprediction of in AdS/CFT the Correspondence, conformally verifiedbulk in flat metric the [33 class – “rigi 37], (1.1). relating correctionstrace deformation to It the partition functional determinant of the GJMS conformal Laplacians.our present T computation is that whenM restricted to the confo free field computation on the boundaryLaplacians involves a on combinatio forms, depending on thesions field content the of coefficient the of mul theholographic Euler and density the is free no CFT longerMoreover, anomaly protect the computation is found on same is true for the coefficients of the Euler and Weyl terms in f infinity. Our working formula willthat then be was the shown natural to genera be valid in dimensional regularization in t at The determinants of the positive Laplacian on the bulk scription. In consequence, there seemsinvolving to the be classical little hope SUGRAoperators to action, (e.g. g conformal the Laplacian) anomaly in associat generic even dimen N function method, involving the resolvent at perbolic space the task is then reducedflatness to of volume the renormalization, boundary with metric.back the to We focus the on holographic the anomalylarization conformal of of ano the the volume renormalized produces volume. andiffers anomaly from and those a obtained correspon byversal standard zeta-regularizatio tional determinant identity comingdouble-trace from deformations the of generalized thecase Ad determinants boundary in conformal the theor lightthe of conjectured several equality possible between regulari Poincare functional metric determinants with prescribedsented conformal in infinity. the case Explic of conformally flat boundary metrics for bo JHEP07(2008)103 has 39]. units α [24], as O A action N ), are then both d ints of a RG flow − , and the generating etail encoded in the λ Y ( ary multi-trace opera- tions λ given by the two roots ry. , respectively, and whose = + d at infinity. The modern two different dual CFTs in y conclude by summarizing 2 λ tation on the dual boundary ne may fix either the faster e transformation at leading ly as well as the renormalized n on the bulk scalar relating m s first computed in the bulk of with “tachyonic” mass in the eported Polyakov formulas for lassical supergravity (SUGRA) maly at leading large of this section we briefly survey s to a double-trace deformation eory (with prescribed boundary theory in the IR, which now has ions. Some background material and tional prescription [20, 21] entail own with the help of dimensional φ − CFT, where the operator − β λ − α limit of the CFT. At this level, there is N of the 2 α , for some compact space – 5 – Y × f O +1 d (1) correction to the conformal anomaly under a flow O ) of the AdS/CFT relation AdS 2 at the conformal boundary. One of the most remarkable m d + 2 4 +1, first considered long ago by Breitenlohner and Freedman [ d 2 4 theory in the UV flows into the CF T d q − − α = < ν ), implying that quantum corrections to this classical SUGR of the gauge group measures the size of the geometry in Planck is dual to an operators of dimension , concerning an 2 4 / Y φ 1 m N . The N ≤ (with − 2 λ 4 = d ν − ± P 2 d /l The rank The generalized AdS/CFT prescription to incorporate bound = AdS ± L functional of the dual successes of this duality is the mapping of the conformal ano an outgrowth of the IR-UV connection [23],action that in relates the the c bulk to a quantum one-loop anomaly on the bounda dimension ( above the unitarity bound. tors [41] provides alinearly dynamical the picture: faster falloff a partof boundary to the conditio the CFT slower one Lagrangian.triggered correspond by The the two relevant CFTs perturbation of above are then the end po GJMS operators in theour literature, holographic to findings our and hint knowledge. atis possible We collected further finall extens in three appendices. 2. The physical motivation: AdS/CFT correspondence The celebrated Maldacena’s conjecture [19]the and equality its between calcula the partitionconditions) function in of the String/M-th product space conformal variations. For comparison, we collect then all r correspond to subleading terms in the large Two AdS-invariant quantizations areor known slower to falloff exist, of sinceAdS/CFT the o interpretation [40] quantum assigns fluctuationswhich the of the same field the bulk scalar theory to fiel compact space produced by a double-trace deformation.AdS This [33] and correction confirmed wa shortly aftertheory by [34] a (see field also [35, theoretic 36]). compu regularization Full in agreement [37], was where finally we sh werevalues able of to the match functional the determinants anoma these involved. preliminary In results. the rest 2.1 The generalized prescription forA double-trace subtle deforma example ofwindow the duality involves a scalar field a universal AdS/CFT result, not relying on SUSY or any other d generating functionals arelarge related N. to The each conformal otherλ dimensions by of Legendr the dual CFT operators, JHEP07(2008)103 2 e N and (2.1) (2.2) − α λ O G 4, see e.g. [42, 43]. (1) contribution to l action. Using the O 4 − s – 9 – ) − − =1 ∞ s l X dx ( λ = 1 0 The zeta regularization produces directly a renor- α 0 Re 2 b Z ∆ X λ> − log det∆ = − det det∆  )= · s log det∆ = ( 3 log ∆ the eigenvalues are also rescaled L ζ comes from the Q-curvature term 2 g of Branson and Ørsted [46], and the remaining 2 α α 2 3 2 e − = b g → g conformal index However, there is still a missing contribution (a global ter The corresponding extension of this computation to higher d must consider: that the above renders an additional contribution 2 In dimensional regularization, the sum over degeneracies s anomaly corresponding to the renormalized volume is known t zero mode, as on the coefficient, we find again the same result as above. cisely the only the term proportional to the renormalized volume in terms of the zeta function on the two-sphere is thrown away in the renormalization prescription. In all, curvature invariants to the integrated conformal anomaly i the global term associated tocan the therefore zero read mode. the From coefficientglobal the of term. rigid the co We get Q-curvature no term information, (ty however, on the local inv whole family of GJMS operators is straightforwardBoundary: in dimens zeta-regularization. malized determinant formula because it is scale invariant and, therefore, under Alternatively, one can use the fact that in the renormalized This totally agrees with the Polyakov formula evaluated for The first contribution The above representation is valid for accomplished by rewriting in termsHurwitz of zeta better functions. studied The zeta result fu in the present case can be JHEP07(2008)103 is in to ǫ 3 c 3 2 l (3.9) (3.12) (3.13) (3.11) (3.10) − +1) = l (2 me role as er-Kinkelin =1 c l l (0) = ∆ P ζ . · finite eft are the expected + + 1) and the IR-cutoff but there seems to be , the counting function c ttacked with methods  l . c l act proportional to the ∆ 2 c ; however, if one instead l tion under rigid rescaling 3 3]: the IR-UV connection (1)) rization nor in heat kernel 4 1 o − he higher GJMS operators. ǫ formula) that grows as xtra log-divergencies is still 0, the heat kernel produces − + om per unit Planck area. ly and explicit results, to our c l onality factor, 2 + 1) log( A → dim ker ( c 2 l δ / log − 2  n in rough agreement with the volume  t − + (2 4 3 6 δ e l R α . √ + 12 4 3 / c log l ∼ 1+ l =  ǫ 2+1 c + 4 l =1 b ∆ n/ l X 2 c Finally, we want to present yet another regular- – 10 – l 1 between the UV-cutoff dvol 2+ finite, 2 det det∆ / 2 ∼ 2  S + n ǫ in the sum over eigenvalues. It is physically appealing Z log n δ 1 · + c 1). The total contribution for the regularized determi- l c 7 3 l πt 1 = − 4 + 1)] = 4 log ( n ′ l  2 3 ( ζ l (0). The input we need is the zeta function 1) + t − + dt A ...n − δ δ ( 1 3 1 12 ′ Z αζ 2 3 ζ 2 2 4 − − = 2 − 1 − + 1) log [ = A 1 = l (2 A c l =1 the spherical modes, then the number of modes plays now the sa l X asymptotics of the remaining sum is determined by the Glaish c l log det∆ = given by c log det l , where log A ). We can subtract them so that the two residual divergences l − c l b A + →∞ Extensions of this computation to higher dimensions can be a We want to read now the integrated anomaly from the log-term, To compute log det∆ we have to consider the finite sum 2 c l n ( · finally get nant is then The integrated anomaly can be read in this case from the varia log det similar to those ofknowledge, [48]. do not The cover the calculations Paneitz are operator ratherBoundary: or length any large-eigenvalue other cutoff. of t ization tool which is simply a cutoff new divergent terms which haveasymptotics. no For analog example, in using the a volume “proper-time” regula cutoff asymptotics. Althoughobscure a to physical interpretation us,counting of their function with these structure the order e is of the simple operator as enough. proporti They are in f and IR-UV connection relates the cutoffs as 2 due to its role regarding the holographic bound in AdS/CFT [2 forces the number of cells in the coarse-grained sphere to be constant as The large enforced by the requirement of (roughly) one degree of freed the number of cells(whose did asymptotics before. is in The general number given of by Weyl’s modes asymptotic is given by truncates at this case. In all, the identification JHEP07(2008)103 + d (4.1) (3.14) (3.15) , i.e. the bulk whose boundary M 1) piece. A com- ]) as its conformal that has a double g − X ( ′ ; [ X ζ ] on esponding asymptotic ional determinants in- 4 . g M x the rigidity, we must on luated for the standard ) − e [ y) adapted to the higher- λ ndary ndence in terms of a ( + . g M g subleading large-N term on nstein manifold. This defines erms in the effective potential, S gularized volume will correctly (1) 1) which is common to the two r leads to the following relation sky constants [49](see also [50]) o e correct boundary result, is still The cutoff regularization of the − g the Green’s function method by ( + ′ ζ ǫ 1 4 with negative cosmological constant, log det ribed by the somehow less rigorous notion of − , log X 7 − 7 3 π 2 = with a metric − )] )] 1) + X 2 λ λ 1. In the present case, the integrated anomaly − ǫ π ( 2 – 11 – ′ − − ∼ ζ d d 4 ( ( δ )= λ λ − √ } · − − c and the renormalized determinant, given by l X X c r>ǫ l [∆ [∆ { − + Vol( det det log is also asymptotically hyperbolic. + and whose interior points are g +1 dimensional manifold with a Poincare metric and ( , so that the volume asymptotics is given by d 3 M H ) as a + The above procedure can be (straightforwardly but tediousl We are then naturally led to the following guess for the funct g In the physics context, this generalization is usually desc ; dimensional (asymptotically) Einstein manifold 7 shows up as coefficient of log − X 3 4 that has a compactification consisting of a manifold with bou 1 pensation between divergences in the volumethat and in vanishing dimensional t regularization conspiredmissing to produce here. th 4. The general case of PoincareAfter metrics this preamble, letfirst us consider now a turn generalization to the [21] main of theme. the AdS/CFT To rela Correspo asymptotically (Euclidean) anti-de Sitter metrics. ones in view of the identification metric is that of( a conformally compact (asymptotically) Ei correctly reproduces the ‘most transcendental’ part When multiplied by the effective potential,reproduce the Hadamard-re the anomaly but not the finite remnant, not even the points are Alternatively, the evaluation of the bulktaking determinant usin the derivative with respect to the spectral paramete previous regularization alternatives. dimensional spheres andestimates to are the determined by whole thein family Glaisher-Kinkelin-Bender this of case. GJMS. TheBulk: corr Hadamard regularizationvolume of [24, the 18]metric volume. produces of a null renormalized volume when eva pole near the boundary so that it defines a conformal structur volved in the one-loopthe bulk boundary: correction and the correspondin infinity. Such JHEP07(2008)103 , +1 is a kov d (4.2) (5.2) (5.1) H , we get ν and consider the . t imaginary argu- d  ) ) λ . λ R-divergent bulk vol- ( k − etric to be conformally 2 M ute the variation under d 2) to make contact with P S ( 51], theo. 1.2) for certain ifficult to compute in gen- , ll next show. X metry is still (isometric to) o the homogeneity of d es will be present. However, happens to be odd and the ) then be rather interested in dλ sumed in the above guess, so ntegrating back in R ν ) d ly computed. In addition, the ( ric for even λ d atio of functional determinants pseudo-differential operator ( on where the bulk computation log det onal determinants of the GJMS explicitly known in terms of the on; they only fail to account for ,...,d/ 1 A c 2 1 2 − M , S −  = ) = 2 = 1 + k g x,x , )] = tr k vol λ d )] ( λ → − +1 d – 12 – − d ν H ( d Z ( , and additional conformally invariant terms with a X · X R +1  d ) R − ν V ( ) − which play no role in the analysis of the conformal varia- d λ ) ( k A λ ) and its analytic continuation ( X ν λ X 2 R ( R X [ dν , which are better known, and with their corresponding Polya R ) tr [ k k 2 0 λ Z 2 P  − d ( ), essentially the Plancherel measure on hyperbolic space a ν ( ) d ( A Unfortunately, the above functional determinants are too d A renormalized version (in DR) leaves a finite remnant of the I continuation in the spectral parameter ( in term of the resolvent An analog relation hasgeneralized been variants shown of to the be trace, valid by but Guillarmou for ([ the case in which tion. The generalized Polyakov formula relating the functi functional determinants are conformal invariants. eral. Only in verydeterminant symmetric of situations the scattering they operator can aslargely be than unexplored of explicit an object. ellipti Yet,at Polyakov formulas conformally for related the metrics r conformally capture invariant valuable terms. informati To makethe variation some under progress, conformal we rescaling will of the boundary met ume, i.e. the renormalized volume non-polynomial dependence in the GJMS operators formulas. Conformal flatness of thethat boundary under conformal metric rescaling is both not typein as A this and paper type B we anomali willwill restrict be to reduced the to conformally that flat of situati the volume renormalization as we5. wi Conformally flat class and volume renormalization To read the (infinitesimal)a anomaly, Weyl one rescaling has ofrelated to the to boundary be the metric. able standard one tothe We on hyperbolic comp let the space the sphere, [22, boundary so 38].hypergeometric m that In function the this and bulk case, one geo the easily resolvent gets is with ment, as in (C.2). There is no dependence on the position due t for the “bare” determinant and therefore the volume factorizes when taking the trace. I JHEP07(2008)103 as a (5.4) (5.5) (5.7) (5.8) (5.3) (5.6) of the , up to +1 ) d d d ( V v 2)! Pff to the d/ ( , encoded in the 2 Pff. We find then k 2 2 d/ . P d/ 2)!  2) 2) d/ − ( − +1 ( d alized volume metric, i.e. it suffices to to work out the type A c formula [26] the relation between the rnative derivation). The = ( e infinitesimal variation of = in even dimension om a generic gravitational log det −V ) d y the coefficient 2 1 d g [37] for the rigid case, to get , ( E w − v of the renormalized volume +1 g . . 2 g the volume renormalization. d perbolic space. We can now use e d b V dv . )  Q 2)! = ) d , · d · d ( ( b g d/  E  v ( ) ) · · · . Up to total derivative terms, we can d w v ν ν g 2 ( ( + ]= d · d M ) g Z [  A d A ( ) V ν ν v ν = 2 2 ( d µν = =0 – 13 – δ ε dν dν A | δg Q k k ] ν 0 2 0 g 2 µν Z Z and the ellipsis stands for derivative terms involving d/ g   εw c dν 2 1 g 2 2 e 2 k = − [ √ d/ 0 c V k k Z 2 2 1)!] 2  b d P P dε − k det det )!( k log ( . The universal Type A anomaly of coefficient in the conformally flat case ) k 2 1 k 2 ) ( d − v [2 ( k v 1) − = ( k c Pfaffian, according to thePfaffian and conventions the of [26] and further use Q-curvature term or in the Euler term, is finally given by conformal Laplacians at conformally related metrics i.e. a trace anomaly functional of the boundary metric representative To get the Q-curvature term we make then use of the holographi 5.2 Type A holographic anomaly In an independentholographic development, anomaly, the coefficient authors ofaction of the which [52] Euler admits were term, able input AdS coming needed as fr is solution theexamine (see the Lagrangian also rigid density situation [53] (in evaluatedthis for Euclidean general for an signature) result the for alte combined hy AdS with the one-loop computation Now, keeping track on the normalizations we can translate volume expansion the global term, is proportional to the conformal variation full agreement with the previous result (5.7) obtained usin the Hadamard-renormalized volume (see e.g. [18]) is given b rely on the well known results obtained via a radial cutoff. Th 5.1 The infinitesimal variation: conformal anomaly We have now to deal with the conformal variation of the renorm where lower coefficients JHEP07(2008)103 . g ) as d ( dv v ) l cur- +1 (5.9a) (5.10) (5.9b) d w d and P −V d 2 1 , , Q 2 d +1 ) + d  ν b g V d − Q or the conformal -term, which are  ( ( g 2 d F F dv w ) ν − ( M iation of the functional F dv ˆ g R e will see, and only the 4], besides being simpler − dν ession for ˆ k g b MS Laplacians, all of them la; this has been explicitly 0 F dv the functional determinant. ewritten in the basis of the d volume and scattering the- plicitly derived and not only Z s, respectively, have been ob- e infinitesimal anomaly. This  normalized volume. A finite al determinant at conformally h between zeta-regularization l part wn Polyakov formulas are due ants that enter in the formulas b fficient as in the conventional he conformal mode on the con- F dv heme that renders the Einstein- herein), whose main motivation M hic formula relating 2)!  Z d/ M + Z g 2 1)! ( + dv d/  − 1)  g d − w ( dv ( d 2 d P d/ Q ) 2 1 – 14 – π + + ˆ (4 g d dv = Q d  b  ) Q ν  w ( d w M A Z M ν ) Z . It contains the universal part of above and additional loca 2 ) ) d d,k ( ( dν d,k v c ( k c 0 Z = 2 =  2 k k 2 2 d/ b P P c det ) given as in (C.2), and the curvature invariants in the det = 2 ν ( ) d log d,k A ( − c where Alternatively, via the connection between the renormalize We get then, up to the global term, for the finite conformal var regularization-scheme dependent, enterzeta-regularized with determinants. different coe Thereon is the clearly boundary a anduniversal Q-curvature Hadamard mismatc term regularization is correctly in reproduced.Hilbert the Any action bulk, sc finite as w (see(renormalized) bulk e.g. action [30, induces 55]) anformal effective is boundary action associated for which t toshown gives in a essentially [56] re the and [57]tained. Polyakov where formu The the advantage Liouville of andand the Riegert compact, result action is by that Chang,their the Qing finite local and variation curvature Yang under [5 invariants can conformal be rescaling. ex 6. Comparison to “experiment” Here we collect allcomputed explicit via results standard we zeta-regularization. areto Most aware Branson of of and the collaborators for kno (see thewas e.g. GJ related [13] to and references sharpThe t inequalities particular values and of extremal theare problems coefficients extracted and for the from local theQ-curvature, invari the relevant Weyl heat invariants plus invariants, a which total are derivative. r in Polyakov formulas. ory [25] Chang, Qing and Yang [54] have found an explicit expr related metrics, we have towe find the can conformal readily primitive ofprimitive do th of in the two Q-curvature term, ways. which gives We the can universa apply a result by Branson f 5.3 Conformal primitive: Polyakov formulas To obtain Polyakov formulas for the quotient of the function with vature invariant terms to correctly reproduce the holograp conformal primitive of determinant JHEP07(2008)103 w is ) ) on g g (ˆ ( 1) (6.3) (6.2) (6.4) (6.1) − ) R , d out by vol vol g . 2(  ) ) dv . g  log g g g ) 2 = ( (ˆ | g dv − dv ing Polyakov J J 2 Polyakov for- dv  2 vol vol J 2  | J 8 g − , V log − |∇ − | ˆ g ) ) dv ˆ g as well. ˆ − g J g g 2 ( (ˆ dv  ) ) − dv J  dv alar which is half the g g 2 Jg ˆ g g 2 2 ( (ˆ ˆ 2 | J vol vol − − ˆ − ˆ J ) (see e.g. [59]), then re- dv  J d dv computed by Gilkey [59], ˆ g 2 vol vol  2 ˆ Ric | Y nsions was anticipated by log ∇ 6 nput needed was computed | ( J M dv ˆ a V ( 4 M Z | log = 2 he Q-curvature as well as the − a − t Z ˆ ˆ J ˆ g M J 1 ( V 90 g Z  − 1 a to finally write: 90 M dv e two-dimensional trace anomaly: ville’s form”. The celebrated non-local 2 Z dv −  M ˆ − J g g Z   32  dv g ) + 13 w 32 − M . g ) d R dv Z  ∆ g − is the Paneitz operator and dv + 2 1 Q dv  gQ 32 6 dv P g ˆ g 3 is obtained after eliminating the conformal factor √ P w Q − + + J d g 2 1 − ˆ g P + − R d b – 15 – dv Q dv R dv ˆ ˆ g g b + Q    ˆ g + b dv dv Q dv Q w w 3 ˆ g √ 6  ˆ J  b P w Q = ( M M w ( 2 1 w Z Z M b w Q dv w M 2), and the conformal Laplacian or Yamabe operator is Z d + π π  M Z 1 1 M Q Z = ∆). g P 24 12 w R/ Z  1 √ +34 1 Y 180 90 = = M w = =5 Z = − − M b ∆ J Z 2 b 14 Y Y = = P 28  =  det∆ det This case is the best known in physics, the original 2 b 2 Y Y det det 2 The result for the Yamabe operator was first obtained by Brans π The six-dimensional case for the Yamabe operator was worked π 1 Q log For the Paneitz operator in four dimensions, the correspond 1 log det det − 720 720 3 ) log = = π 2 ) b 2 P P π is the four-dimensional Q-curvature, 7! (4 (4 · Q det det 2 1 3 In two dimensions the Q-curvature is given by the Schouten sc − − These formulas ought to be called Polyakov formulas in “Liou log 8 − Yamabe, d=6. covariant form in terms of the Green’s function of where Branson [13]. The starting point is the heat kernel coefficien Laplacian, d=2. Here the local invariant involves now the Schouten tensor formula was obtained by Bransonfrom [60] Gilkey’s and work the [61], heat invariant i by solving the conformal relation restricted to the conformally flat case. strict to the conformally flatlocal class curvature to invariants entering read in the the coefficient Polyakov of formul t the Schouten scalar. Paneitz, d=4. Yamabe, d=4. mula [8] for the effective action (conformal primitive) of th simply the Laplacian ( and Ørsted [58],Riegert although [7]. the The general input structure needed in is the four heat dime kernel coefficient scalar curvature ( JHEP07(2008)103 (6.8) (6.6) (6.7) (6.5) ed out ) of the d of the Q- 9 ) of its kernel, . by the Bernoulli A [46] restricted to for the Polyakov , ( n q ) · · · 1) B lt for the conformal 2 +1 d, 2 +1 , d ( 4 d )+ can make a longer list c π g 2 ≥ ly, the coefficient of the Γ( dv d omputed from Avramidi’s + 1)] ) 8 c manipulations d i ) := sion. ( or Q d arization. Once we know the sphere. They generalize early 263 i = 10 ions. Remarkably, there is a ( the Q-curvature term and valuable 7484400 ) Γ( d , + the powers Y − he computation becomes almost d c g ˆ g t ( [ 2 Y dv dv c − d d 8 2 =0 S i Y 23 b Q Q = 8 d/ . There is a vast literature computing ( 113400 )= d A M d w dt − Z conformal index theorem S ) ( l (0) on c M B Z Y ζ V ol (1+ 1 = 6 )= 756 – 16 – ) B d d A 0 ( 23 Z q ) Γ( 1360800 1)! d 1 90 ( Table 1: = 4 − 1 − Y − (0) + (0) on the sphere and the dimension d c = d A A ( ζ ζ b Y Y − )= (0) Y ( det det Y q ζ (0) = log Y 4 ) is 1 in two dimensions and zero for all other even dimensions ζ ) (0) + The eight-dimensional case for the Yamabe operator was work Y 2 π ( Y is simply the volume of the sphere times the constant value Γ( q (4 ζ d − = 1) can be easily compared to the above formula, the integral S k . This produces the results in table 1. n is the order of the differential operator B l We find agreement with all these values. Our holographic resu In this case, I am indebted to L. J. Peterson for providing the coefficient of 9 the zeta function for theresults (conformal) Laplacian by on Weisberger the round [47]compact and recipe obtained are by rather Cappelli lengthly and D’Appollonio calculat [64] f explanations. so that we can check the coefficient of the Q-curvature where 2 Yamabe, d=8. where, after performing the integral, one has to substitute we can work outQ-curvature. the The coefficient main relation ofthe is the conformally given Euler by flat term the class: or, equivalent by Branson and Petersonresult [62], [63] where for the the necessary relevantprohibiting input heat and was invariant. it c At was this only stage, done t with computer aided symboli curvature on formula for the conformal Laplacian (5.10) in any evenLaplacian dimen ( number 6.1 Further data for theAs conformal far Laplacian as webased are on interested explicit in computations thezeta-function on type the of spheres A the via anomaly operator zeta coefficient, regul we JHEP07(2008)103 (6.9) 9 / . - - - - ] = 5 2 k ν − 4016750 2 i [ 1 9 15 / − 2 / =0 ormally flat boundary i Y d/ l index an even simpler - - - = 4 etween functional deter- renormalization correctly e bulk. The holographic erators, in agreement with k dν 75776 rtant test of the AdS/CFT of the renormalized volume p to regularization-scheme that appears in the volume nomaly in even dimensions, onformal invariant obtained erivation of Polyakov formu- 1 − for the conformal Laplacian, 253184 l in the bulk and subleading 0 ormalization. This makes, on rvature term in the Polyakov ric formula (5.10) for the type fficients of the additional local he conformal Laplacian, which Z FT and the conjectured equal- -regularization on the compact 2 ζ , we have gone further and shown 15 d/ ) / ! 1) d d,k - - ( = 3 − 738 c 1026 k · 1944 2( 2 − = d/ ) 2 d π ) in Polyakov formulas for GJMS operators ν (4 are, on the other hand, similar to those for the ) 9 X – 17 – − · 3 / ( L 15 d,k / ( 2 d / 3 - c ) 64 / ν 1984 − ( 832 + 1)! − 112 = 2, Paneitz d dν ( k 1 0 Z 2 X d/ ! X X X 1) d X − 9 15 3 3 / / / 2( / 2 4 10 526 184 − Q-curvature coefficient = 1, Yamabe ]= − k Y [ q = 2 = 4 = 6 = 8 Table 2: = 10 (0) + d d d d d Y ζ Graham [18] already noticed that the invariance properties are reminiscent of those for the functional determinant of t formula Q-curvature on the round sphere. We get then for the conforma 7. Conclusion The main thrust of this paperlas has for been GJMS towards a operators. holographic d correspondence On one in hand, this the constitutesdescription general an of impo case double-trace deformations of ofity a the between boundary Poincare partition C metric functionlarge-N in at order the th one-loop on quantum theminants. leve boundary We lead have beenmetrics, to reducing able a the to bulk remarkable computation make identitythe to progress b that other of in hand, volume a the ren renormalization direct case of connection of Poincare between conf metrics theformulas and Q-curvature for the conformally covariant universal operators. Q-cu A We conformal get anomaly a associated to gene thesome whole previously family known of results GJMS op that we summarize in table 2. V constant term in the expansion of the integrated heat kernel which vanishes in odd dimensionsby but integrating in a even local dimensions expression is inthat a curvature. c the In this above note dependent similarities terms. can The bereproduce Polyakov promoted the formulas universal to obtained Q-curvature term. via equalities,curvature volume However, invariants u the are coe different from those obtained via is conformally invariant in oddand dimensions that but the which properties has of an the a invariant JHEP07(2008)103 (7.1) = 6 in [67] d - - - - = 5 299376 283309 k − formula for the zeta , ty, to describe the type e dimension. Here, the 2 d - - - 592 = 4 cki residue (see e.g. [65]) ) urely QFT considerations 467775 ndered Euler density” has ν e results; one possible way 562603 604800 k e of Kleinian groups, where oing beyond conformal flat- , the Q-curvature certainly − − aw under conformal rescaling ation that corresponds to the geometry. ( ature and predicts new ones aturally arise (see e.g. [71, 72] d sharp inequalities [13] may cit expression for ositivity of the induced action efore it is nothing but the Q- nt years. On the physical side, e AdS Schwarzschild black hole and 2 d direction, the relation between with the topology of the confor- ) 1 ν - - 420 379 ( = 3 1400 19 d 92400 − k − S dν oop effect corresponding to the double-trace Even the rigid case of hyperbolic space k 0 Z 10 2 (0) on k X d/ 2 ! P 4 d 31 1) ζ - 945 13 467775 − – 18 – 45 38 28350 − ( − − 2 have led Anselmi [66, 67] to introduce a “pondered k = 2, Paneitz k = k Table 3: 2 d, δ X X in six dimensions [68]. X a-theorem X X 6 (0) + Q 2 1 3 k 90 23 1 2 263 756 − P 113400 − ζ 7484400 = 1, Yamabe − k = 8 = 2 = 4 = 6 = 10 d d d d d Notwithstanding, we can unambiguously write down a compact We have favored the Q-curvature, rather than the Euler densi There are still several interesting issues to be explored. G In a celebrated example due to Hawking and Page [69], where th 10 -regularization on the boundary. thermal AdS share thedeformation same has conformal already infinity, the been bulk explored one-l in [70]. function of the GJMS operators on the round sphere boundary. It thus remains aζ challenge to find a bulk regulariz which correctly reproduces(table all 3). values It reported wouldgoes in be via desirable the the to liter relationand have of the a computation the confirmation of of conformal the thes anomaly later with using symbol the calculus. Wodzi A anomaly. This is mainlyand due to its its simpler simpler transformationplays conformal l a primitive. central role and Init has conformal has been been geometry intensively studied lessregarding in explored; the rece however, irreversibility of let thefor us RG the mention conformal flow, that factor unitarity some and and p p coincides with Branson’s can be extended to quotientsconnections by with symmetry number groups, theory as via isand Selberg the [73], zeta cas functions section2.9, n forPolyakov a formulas, related extremal discussion). ofwell functional In admit a another determinants holographic an interpretation in terms of the bulk Euler density” in thea study linear of transformation conformal lawcurvature anomalies. modulo under Weyl conformal This invariant terms. recaling, “po Moreover, ther the expli ness will switch onissue the of uniqueness Weyl-terms of whose themal number filling boundary, grows Poincare will metric, surely with together play th an important role. JHEP07(2008)103 (A.3) (A.1) ) and 11 ), confor- M positive Lapla- ( ∞ C e the ∈ σ , d g lower order terms f Differential Geometry σ Juhl. I am also indebted n 2 2 of the kernel ry, let us go back to the e ) (A.2) + ( ble advice in preparing this l discussions and references. agement. ncare metric associated to a 1 x −→ = el reduces to that of the k-th − ( ) k bg d ( λ > d/ in a d-dim Riemannian manifold δ k =∆ . ∆ k S k 1)! c . ) − λ − f 2 built using the Fefferman-Graham ambient | σ k , since in the neighborhood of these values 1 = k k x 2 ! ( N and the Riemann tensor (actually the Ricci −→ 2 2 k − k , where 1 P d ∈ − d k k ∇ − e – 19 – ′ 2 λ ( Q 2 2 2 k | x −→ k 2 | k, k 2 , the conformally invariant operators of GJMS [9]. x d/ 2 = P −→ k − | ) σ 2 d k − k c P + 2+ 2 λ +2 Z d k )+ d/ 6= − ∇ 2+ e = k 2 d/ S lim = λ ( d/ , a conformal invariant (covariant) differential operator. → k k f λ − k 2 hanging around, just because in the mathematical literatur ∇ · k b P The GJMS operators k =∆ ), under a conformal change of metric 1) k 2 lower order terms − and M P ( , N + ( ∞ d k ∈ R C k ∈ ∃ has a polynomial expansion in has the form is self-adjoint =∆ is a local scalar invariant. f k k k k k d k 2 2 2 2 2 tensor) in which all coefficients are rational in the dimensio P P Q P On flat P mal covariance: for P ) The generalization of this observation [25] for a filling Poi is preferred. There is a factor ( • • • • • • • 11 M, g cian Acknowledgments My deep gratitude to Harald Dorn for encouragement and valua manuscript. I benefitedand from Global common Analysis seminars at with HU-Berlinto the given H. by Baum, group J. G. o Jorjadze, Erdmenger L.J.Finally, and Peterson I A. and thank H.S. the Yang for Floro Alumni usefu P´erez Forum for theA. encour GJMS operators and Q-curvature To give a glimpsePoincare patch of and these examine the constructions analytic in continuation to conformal geomet given conformal structure involves where Therefore for these “resonantpower values” of the the action Laplacian of ∆ the kern It will have single poles at GJMS operators. construction have, among others, the( following properties (see e.g. [74]) JHEP07(2008)103 (B.1) (A.4) (A.9) (A.6) (A.5) (A.7) (A.8) (A.10) k k 3 2 σ +2 − d d ≥ = 0) to write the u d , 2+1) r k σ d 2 d/ ( 1) . based on the GJMS d Q σ − ) 2 b J e k riance of its volume inte- ·∇ ptotically) Einstein man- d/ in higher dimension and ( ture at ary (at 1) . − e σ 2 2 he Fefferman-Graham theo- − the Yamabe equation +2 − ∇ ∆ d d prescribed Gaussian curvature 2 d/ σ σ 2) 1) 1) b d/ J u − − − = ( 2 2 d 1) . = ( d/ d/ } u ( ( r − 1)( e σ. k − σ g d 2 2 e − 1) P σ , Q + := 2 − d/ d ) 2 2 ( +∆ + u 2 k − Yamabe equation d/ dr J ( − = ( Q d − { σ 2 u 2 = and J e = = – 20 – ] − r J b d/ σ J 1) 1) b Q 1)∆ 1) renders the higher (even-)dimensional generalization − σ = R d − 2 generalizes in many ways the 2-dim scalar curvature dσ − ·∇ k e − e + 2( 2 2 d σ g 1) + ( 2 d/ ∇ := → − d/ 1) + 2( d J u ( − R 2 that results in the PGC eq. . The bulk geometry can be partially reconstructed by an = 2. . 1) + ( ∇ d 2 + curvature d = k [∆ + ( P → − S d/ b d g − ( R σ d := σ − 1) 2 Q − ∇ · − e P 2 σ = d/ , in the limit ∆ ( ) = The σ e ) u + and k g k d d ( − 2 ∆( 2 P Q d/ ( Ric (PGC) in 2-dim starting from the e := = Q u Among the properties of the Q-curvature, the conformal inva Start with the conformal transformation of the scalar curva The very same trick applied now to the higher-order Yamabe eq . It original derivation tries to mimic the derivation of the and absorb the quadratic term where Q-curvature. gral easily follows. B. Q-curvature and volume renormalization Let the bulk metric toifold, be i.e. that of an conformally compact (asym of the PGC eq. with The trick (due to Branson) is now to slip in a 1 to rewrite as analytically continuing to equation R to get for the Schouten scalar asymptotic expansion, whichrem is [10]. essentially One the can always content find of local coordinates t near the bound bulk metric as and take now the limit operators JHEP07(2008)103 is is for en- h d g enor- (B.2) (B.3) (B.4) (B.6) (B.5) 2, w 2 − e d are tensors → ≤ is no longer ) g j j ( V g ≤ being the round , , · · · d 0 + dr g +1 are locally formally g r d r ) even 0 dv j ( j t terms in volume renor- log g d coefficient in the volume with 4 oundary, the invariant one ···} 1, hr · · · ) 0 e boundary metric gives rise d g + ( ly given by the “holographic + + − sz regularizations 2 gulator is removed. When ein equations. v is even, in turn, d ) -term in the Hadamard (cutoff) d d 2 d r ǫ r r ) d r ) ) d d d ( . ≤ ( ( − v g g g is formally undetermined, subject to , j ) is locally determined, but its trace-free g dv d )+ ( )+ )+ ) d ≤ dv d = (1 g of the log ( Q ) g r d ( L g M v Z in the Hadamard regularization scheme. 2 M g – 21 – Z is odd. d/ c are locally determined for dv d = ) ) even powers j d even powers even powers ( ( L are locally determined in term of curvature invariants = 2 g dr + ( +1 , + ( + ( g d L w v 2 d 2 2 happens to be connected to this invariant: r =0 if r dv r r R ...d ) r V d (2) ( + (2) (2) v v g g is trace-free. For 0 = 1 detg detg + + ) in the expansion (renormalized volume) turns out to be indep integral, a regularization is needed. Then a subtraction (r . It is the coefficient d 1+ , j ( ) . The “reconstruction” theorem leads to the asymptotics − { s V g (0) (0) j d r ( V→V g g S v = = = = corresponds to the choice + g r r even : g odd : g +1 dv makes is the chosen metric at the conformal boundary. For odd d d d g w = AdS (0) g conformal anomaly Before taking the The volume element has then an asymptotic expansion The Q-curvature enters here and provides one of the importan metric on the sphere where Euclidean where all coefficients determined by the conformal representative but of the boundary metric and on the boundary and locally determined and trace-free. The trace of the trace-free condition. For even invariant and its variation underto a the Weyl transformaton of th malization) prescription renders a finite answer when the re dent of the conformal choice of the boundary metric. If regularization or residue at the pole in dimensional and Rie odd, the finite remnant part is formally undetermined. All this is dictated by Einst expansion, up toformula” total-derivative [26]. terms which are explicit Therefore, the Q-curvature is then proportional to the malization asymptotics at conformal infinity [25] infinitesimal given by the integralin of this a case. local The curvature variation expression of on the b JHEP07(2008)103 (C.5) (C.4) (C.7) (C.2) (C.6) (C.1) (C.3) ). , a . +1 Γ( d / ) b ·L  (cf. eq.2.13 in [13]) o the renormalized ν ), which is the two- +  . ) − + a x . λ ( 2 d vol d ) )  d ical harmonics B ν ν alization of the ball model := Γ( sidue of the pole term ψ and x one-loop effective potentials +1 − + 2 b d tance divergences cancel out ) − , expanding in spherical har- − 2 2 d d H intertwiner , a d 2 d λ Z dx l  S ) + + · ν ν ν l l 1) 0 2 d  ! ) − Z ) + l Γ( Γ( − ( x  1 2 ( 2 d 2 d ( d d ( , )  + ) ν A 1 ) log ( ψ ν x ( − 1 +1 2 d 2 d − d, l d d π − 1  dx ·A d ·V + d 2 d ν 2 l deg( – 22 – 0  2 ν ) +1 Z − 1 d 2 =0 ∞  x l 2 1 X ( L d )= =  = A ψ ) = α d, l x ν − grav 2 ( /Z d )+ β /Z dx deg( π A Z ν 0 + grav Z Z  log(4 2 log on the round sphere (see e.g. [75]).  are nothing but the eigenvalues of the = − log i ) ) ) β ν ν ν − ( O 2 + − − grav d β 2 2 d d A + + O /Z l l h Γ( Γ( + grav )= ν Z ( d B log − We extended the mapping from that of the integrated anomaly t The boundary computation on the standard sphere and the renormalized determinant is given by and the ratio of the hyperbolic space with the standard metric: C. Rigid case in dimensional regularization The rigid computation in the bulk involves the volume renorm point function partition functions as well. The anomaly can be read as the re conveniently written in terms of the Pochhammer symbol ( where between conjugate representations (with conformal labels The factor in square bracketsassociated comes from to the the difference two ofto asymptotic the render behaviors, a whose finite result short with dis Here we have a weighted sum with the degeneracies of the spher monics, results in an UV-divergent sum JHEP07(2008)103 , ]. Helv. Proc. ry , , On the ; A 107 . 347 ecture of Osaka J. , (2007) 370 lds sion thereof (1981) 207. (1984) 56. n for the Weyl Phys. Lett. Sharp inequalities, , hep-th/07060340 B 644 n¨uber rbitrary Proc. London Math. Soc. math.DG/0509571 , B 103 B 134 arXiv:0707.1737 , (2006) 466 , ng rea (1993); ]. 206 Conformally invariant powers of Trans. Am. Math. Soc. Elie Cartan et les Math´ematiques Phys. 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